In this paper, we show that Bandwidth is hard for the complexity class $W[t]$ for all $t\in {\bf N}$, even for caterpillars with hair length at most three. As intermediate problem, we introduce the Weighted Path Emulation problem: given a vertex-weighted path $P_N$ and integer $M$, decide if there exists a mapping of the vertices of $P_N$ to a path $P_M$, such that adjacent vertices are mapped to adjacent or equal vertices, and such that the total weight of the image of a vertex from $P_M$ equals an integer $c$. We show that {\sc Weighted Path Emulation}, with $c$ as parameter, is hard for $W[t]$ for all $t\in {\bf N}$, and is strongly NP-complete. We also show that Directed Bandwidth is hard for $W[t]$ for all $t\in {\bf N}$, for directed acyclic graphs whose underlying undirected graph is a caterpillar.
翻译:在本文中,我们显示,对于复杂等级($W$_N$)来说,即使毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛